Theory
======

Mathematical background for quantum information geometry and the inaccessible game.

.. toctree::
   :maxdepth: 2
   
   quantum_exponential_families
   fisher_information
   inaccessible_game
   origin_regularisation
   generic_structure
   symbolic_computation

Quantum Exponential Families
-----------------------------

*Coming soon: Mathematical theory of quantum exponential families*

Topics:
- Definition and properties
- Log-partition function :math:`\psi(\theta)`
- Expectation parameters and duality
- Quantum Fisher information

Fisher Information and the BKM Metric
--------------------------------------

*Coming soon: The Bogoliubov-Kubo-Mori metric*

Topics:
- Definition via perturbation theory
- Kubo-Mori inner product
- Relationship to quantum Fisher information
- Geometric properties

The Inaccessible Game Framework
--------------------------------

*Coming soon: Information-geometric perspective on constrained dynamics*

Topics:
- Marginal entropy constraints
- The inaccessible region
- Information gain vs accessibility
- Relationship to thermodynamics

Origin Regularisation
---------------------

The LME origin (product of Bell states) lies at the pure-state boundary
where natural parameters θ → -∞. The regularisation matrix σ encodes the
"direction of approach" to this boundary. Key topics:

- The north pole analogy (coordinate singularity at pure states)
- Valid σ requirements (Hermitian, PSD, unit trace)
- Efficiency implications (isotropic vs product vs general)
- Block-diagonal Fisher information for product states

See :doc:`origin_regularisation` for full details.

GENERIC Structure
-----------------

*Coming soon: General Equation for Non-Equilibrium Reversible-Irreversible Coupling*

Topics:
- Decomposition of dynamics: :math:`\dot{\theta} = F(\theta) = F_{\text{rev}} + F_{\text{irr}}`
- Jacobian structure: :math:`M = S + A`
- Relationship to Hamiltonian and gradient flows
- Entropy production